login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A114775 Expansion of -x^2*(x^2+1)*(x^7-x^4+1) / ((x^6-x^4+1)*(x^6+x^4-1)). 0
0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 4, 5, 7, 5, 7, 9, 12, 9, 12, 16, 21, 16, 21, 28, 37, 28, 37, 49, 65, 49, 65, 86, 114, 86, 114, 151, 200, 151, 200, 265, 351, 265, 351, 465, 616, 465, 616, 816, 1081, 816, 1081, 1432, 1897, 1432, 1897, 2513 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

Old name: Matrix Markov of a complex sequence based on the Minimal Pisot (A000931) characteristic cubic having its roots rotated by 90 degrees or "I".

This procedure essentially divides the sequence A000931 into two parts as real and imaginary. As such it is possible to plot the result on the complex plane where it gives a square outward spiral: b = Table[{Re[v[n][[1]]], Im[v[n][[1]]]}, {n, 0, 25}] ListPlot[b, PlotJoined -> True]

LINKS

Table of n, a(n) for n=0..65.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1,0,0,0,1).

FORMULA

M = {{0, 1, 0}, {0, 0, 1}, {-I, -1, 0}}; v[0] = {0, 1, 1}; v[n_] := v[n] = M.v[n - 1] {a(n),a(n+1)}= {Abs[Re[v[n][[1]]]], Abs[Im[v[n][[1]]]]}

G.f.: -x^2*(x^2+1)*(x^7-x^4+1) / ((x^6-x^4+1)*(x^6+x^4-1)). [Colin Barker, Jan 01 2013]

EXAMPLE

Derivation:

a[n_] := x /. Solve[x^3 - x - 1 == 0, x][[n]]

FullSimplify[ExpandAll[Product[(x - I*a[n]), {n, 1, 3}]]]

Gaves the polynomial:

x^3+x+I

as does:

Det[M-x*IdentityMatrix[3]]

NSolve[Det[M-x*IdentityMatrix[3]]\[Equal]0,x]

MATHEMATICA

M = {{0, 1, 0}, {0, 0, 1}, {-I, -1, 0}}; v[0] = {0, 1, 1}; v[n_] := v[n] = M.v[n - 1] aout = Flatten[Table[{Abs[Re[v[n][[1]]]], Abs[Im[v[n][[1]]]]}, {n, 0, 25}]]

CROSSREFS

Cf. A000931.

Sequence in context: A265626 A230230 A187821 * A071136 A025425 A234451

Adjacent sequences:  A114772 A114773 A114774 * A114776 A114777 A114778

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Feb 21 2006

EXTENSIONS

New name from Colin Barker, Jan 01 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 17 09:22 EDT 2019. Contains 325100 sequences. (Running on oeis4.)